We use 15 years of $\gamma$-ray data from INTEGRAL/SPI in a refined investigation of the morphology of the Galactic bulge positron annihilation signal. Our spatial analysis confirms that the signal traces the old stellar population in the bulge and reveals for the first time that it traces the boxy bulge and nuclear stellar bulge. Using a 3D smoothing kernel, we find that the signal is smeared out over a characteristic length scale of $150 \pm 50\,$pc, suggesting either annihilation in situ at astrophysical sources kicked at formation or positron propagation away from sources. The former is disfavoured by its requiring kick velocities different between the Galactic nucleus ($\gtrsim 50\,\mathrm{km\,s^{-1}}$) and wider bulge ($\lesssim 15\,\mathrm{km\,s^{-1}}$) source. Positron propagation prior to annihilation can explain the overall phenomenology of the 511 keV signal for positrons injection energies $\lesssim 1.4\,$MeV, suggesting a nucleosynthesis origin.